Jem shows bravery in the very first chapter. Dill says that Jem is too scared to put his toe in the Radley's front yard. Eventually, Dill bets Jem that he can't touch the Radley house. Jem was scared but he overcame that fear and ran up and touched the house.

In Chapter 15, Atticus goes to guard the Maycomb jail where Tom is being held. Sure enough, a mob comes to the jail seeking to harm Tom. When the mob approaches, the children, who've been watching, come to Atticus' aid. Scout is actually the one who initiated the charge, but Jem shows his bravery as well when he refuses to leave. Scout stands out as the brave child in this chapter but she stays because Jem stayed. If Jem had left, she and Dill would have followed. The presence of the children diffuses the intense situation and Scout's words prompt Mr. Cunningham to call off the mob.

Atticus recognized that it was Jem's bravery that changed the tone of the situation and because he stayed, Scout stayed and worked her naive, childish innocence on Mr. Cunningham. Scout thought that Atticus would be furious with Jem but the opposite was the case:

Atticus and Jem were well ahead of us, and I assumed that Atticus was giving him hell for not going home, but I was wrong. As they passed under a streetlight, Atticus reached out and massaged Jem’s hair, his one gesture of affection.

Another significant moment is when Jem learns something about bravery from Mrs. Dubose. As punishment for cutting her camellia bushes, Mrs. Dubose has the children come to her house and read to her. Atticus endorses this punishment for a reason. Although Mrs. Dubose behaves like a mean-spirited old woman, Atticus wanted the children to recognize her bravery in kicking her morphine addiction despite the pain that it caused. Atticus tells Jem:

I wanted you to see what real courage is, instead of getting the idea that courage is a man with a gun in his hand. It’s when you know you’re licked before you begin but you begin anyway and you see it through no matter what. You rarely win, but sometimes you do.